[Proof of Lemma REF ]
Part (a) follows directly from the definition of \( {\overline{G}} \) . Recall that a bivariate function \( {\overline{G}} \)  is positive semi-definite iff for any \((c_1,\ldots ,c_m)\in \mathbb {R}^m\)  and any \(\lbrace r_j\rbrace _{j=1}^m\subset \mathbb {R}^{d}\) , the sum \(\sum _{k=1}^m\sum _{j=1}^m c_kc_j {\overline{G}} (r_k,r_j)\ge 0\)  (see e.g. [1]}, [2]}, [3]}). Then, noting that from (REF ) and (REF ) we have
\(& \sum _{k=1}^m\sum _{j=1}^m c_kc_j {\overline{G}} (r_k,r_j) \\= & \frac{1}{N}\sum _{i=1}^N\int _{|\eta | =1} \int _{|\xi |=1} \left[ \int \sum _{k=1}^m\sum _{j=1}^m c_kc_j\frac{[u_i(x+r_k\xi ) - u_i(x) ] [u_i(x+r_j\eta ) - u_i(x) ]}{\rho _N^{\prime }(r_j)\rho _N^{\prime }(r_k)} dx \right]d\xi d\eta \\= & \frac{1}{N}\sum _{i=1}^N\int _{|\eta | =1} \int _{|\xi |=1} \left[ \int \left|\sum _{k=1}^m c_k\frac{[u_i(x+r_k\xi ) - u_i(x) ] }{\rho _N^{\prime }(r_k)} dx\right|^2 \right]d\xi d\eta \ge 0.\) 
